P. Clarke describes mirror symmetry as a duality between Landau-Ginzburg models, so that the dual of an LG model is another LG model. We describe examples in which the underlying space is a total space of a vector bundle on the projective line, and we show that self-duality occurs in precisely two cases: the cotangent bundle and the resolved conifold.

This paper presents the theory of Bohr-Sommerfeld-Heisenberg quantization of a completely integrable Hamiltonian system in the context of geometric quantization. The theory is illustrated with several examples.

In this paper we study harmonic analysis on the Einstein gyrogroup of the open ball of ${\mathbb R}^n, n \in \mathbb{N},$ centered at the origin and with arbitrary radius $t \in \mathbb{R}^+,$ associated to the generalised Laplace-Beltrami operator $$ L_{\sigma,t} = \left( 1 - \frac{\|x\|^2}{t^2} \right) \!\left( \Delta - \sum_{i,j=1}^n \frac{x_i x_j}{t^2} \frac{\partial^2}{\partial x_i \partial x_j} - \frac{\kappa}{t^2} \sum_{i=1}^n x_i \frac{\partial}{\partial x_i} + \frac{\kappa(2-\kappa)}{4t^2} \right)$$ where $\kappa=n+\sigma$ and $\sigma \in {\mathbb R}$ is an arbitrary parameter. The generalised harmonic analysis for $L_{\sigma,t}$ gives rise to the $(\sigma,t)$-translation, the $(\sigma,t)$- convolution, the $(\sigma,t)$-spherical Fourier transform, the $(\sigma,t)$-Poisson transform, the $(\sigma,t)$-Helgason Fourier transform, its inverse transform and Plancherel's Theorem. In the limit of large $t,$ $t \rightarrow +\infty,$ the resulting hyperbolic harmonic analysis tends to the standard Euclidean harmonic analysis on ${\mathbb R}^n,$ thus unifying hyperbolic and Euclidean harmonic analysis.

The explicit matrix realizations of reversion and spin groups depend on the set of matrices chosen to represent a basis of one-vectors for a Clifford algebra. On the other hand, there are iterative procedures to obtain bases of one-vectors for higher dimensional Clifford algebras, starting from those for lower dimensional ones. For a basis of one-vectors for $\mathrm{Cl}(0,5)$ , obtained by applying such procedures to the Pauli basis for $\mathrm{Cl} (3,0)$ the matrix form of reversion involves neither of the two standard matrices representing the symplectic form. However, by making use of the relation between $4\times 4$ real matrices and the quaternion tensor product ($\mathbb{H}\otimes \mathbb{H}$), the matrix form of reversion for this basis of one-vectors is identified. The corresponding version of the Lie algebra of the spin group, $\mathfrak{spin}(5)$, has useful matrix properties which are explored. Next, the form of reversion for a basis of one-vectors for $\mathrm{Cl}(0,6)$ obtained iteratively from $\mathrm{Cl} (0,0)$ is obtained. This is then applied to computing exponentials of $ 5\times 5$ and $6\times 6$ real antisymmetric matrices in closed form, by reduction to the simpler task of computing exponentials of certain $4\times 4 $ matrices. For the latter purpose closed form expressions for the minimal polynomials of these $4\times 4$ matrices are obtained, without availing of their eigenstructure. Among the byproducts of this work are natural interpretations for members of an orthogonal basis for $M(4,\mathbb{R})$ provided by the isomorphism with $\mathbb{H}\otimes \mathbb{H}$, and a first principles approach to the spin groups in dimensions five and six.

In Euclidean space, the geodesics on a surface of revolution can be characterized by means of Clairaut's theorem, which essentially says that the geodesics are curves of fixed angular momentum. A similar result is known for three dimensional Minkowski space for timelike geodesics on surfaces of revolution about the time axis. Here, we extend this result to consider generalizations of surfaces of revolution to those surfaces generated by any one-parameter subgroup of the Lorentz group. We also observe that the geodesic flow in this case is easily seen to be a completely integrable system, and give the explicit formulae for the timelike geodesics.